The equations of the pair of asymptotes of the hyperbola $x^2-2y^2-8x+8y+4=0$ is 

$x^2-2y^2-8x+8y+4=0\to \left(1\right)$
The equation of pair of asymptotes of (1) is 
$x^2-2y^2-8x+8y+k=0\to \left(2\right)$
(2) rep. a pair of lines 
$\Rightarrow abc+2fgh-a{f}^2-b{y}^2-c{h}^2=0$
$\Rightarrow k=8$
$\left(2\right)\Rightarrow x^2-2y^2-8x+8y+8=0$